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The Empirical FT. What is the large sample distribution of the EFT?. The complex normal. Theorem. Suppose X is stationary mixing, then. Proof. Write. Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments. Consider. Comments .

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slide1

The Empirical FT.

What is the large sample distribution of the EFT?

slide4

Proof. Write

Evaluate first and second-order cumulants

Bound higher cumulants

Normal is determined by its moments

slide6

Comments.

Already used to study rate estimate

Tapering makes

Get asymp independence for different frequencies

The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0

Also get asymp independence if consider separate stretches

p-vector version involves p by p spectral density matrix fXX( )

slide7

Estimation of the (power) spectrum.

An estimate whose limit is a random variable

slide8

Some moments.

The estimate is asymptotically unbiased

Final term drops out if  = 2r/T  0

Best to correct for mean, work with

slide9

Periodogram values are asymptotically independent since dT values are -

independent exponentials

Use to form estimates

slide16

Estimation of finite dimensional .

approximate likelihood (assuming IT values independent exponentials)

slide18

Crossperiodogram.

Smoothed periodogram.

slide22

Large sample distributions.

var log|AT|  [|R|-2 -1]/L

var argAT [|R|-2 -1]/L

slide23

Advantages of frequency domain approach.

techniques formany stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

time varying variant

...

slide24

Networks.

partial spectra - trivariate (M,N,O)

Remove linear time invariant effect of M from N and O and examine coherency of residuals,

slide27

Point process system.

An operation carrying a point process, M, into another, N

N = S[M]

S = {input,mapping, output}

Pr{dN(t)=1|M} = { + a(t-u)dM(u)}dt

System identification: determining the charcteristics of a system from inputs and corresponding outputs

{M(t),N(t);0 t<T}

Like regression vs. bivariate

slide29

Realizable case.

a(u) = 0 u<0

A() is of special form

e.g. N(t) = M(t-)

arg A( ) = -

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